Marxs Method in Social Science, and
its Relationship to Classical and Modern Physics and Mathematics

Johannes Witt‑Hansen
/ Copenhagen

In an essay, "The Scottish Contribution to
Marxist Sociology", Ronald Meek sets out to comment upon certain "attempts
to give the writing of history a materialistic basis" which were made by
a group of eighteenth century Scottish writersthe so-called "Scottish
Historical School". [1] Outstanding members of this group were David Hume
(1711‑1776), Adam Smith's teacher, the historiograph, sociologist and
physicist, Adam Ferguson (1723‑1816), Adam Smith (1723‑1790), the
philosopher and economist, Dugald Stewart (1753‑1828), and the historian
and sociologist, John Millar. They were all, with one exception, viz. John Millar,
quoted by Marx.

Meek's contention is, roughly, that the sociological
work of these writers and their impact on the sociological conception of history
has been seriously underestimated. In his essay Meek gives ample textual evidence
for such point of view, and points out that "their approach (at least in
intention) was "scientific" in the best sense of the word 
as implied in the description of their subject as the "natural history"
of law, government, etc." [2].

There is, however, another interesting aspect of
the relationship between the "Scottish Historical School" and Marxian
social science. This aspect concerns Marx's method in social science,
often called "the dialectic method". [3]

It is, noteworthy that contemporary members of the
Socialist movement and other contemporaries had a very modest understanding
of Marx's strictly scientific work, and that they misjudged or underestimated
the purport of his method in social science. On occasion of the publication
of A Contribution to the Critique of Political Economy (1859), one of
the founders of the German Social Democratic Party, Wilhelm Liebknecht, confessed
to the Socialist German journalist, Elard Biskamp, that "never had a book
disappointed him so much". And Biskamp declared openly that he did
not see either "à quoi bon". [4] In a letter to the German Socialist,
Ludwig Kugelmann, July 11, 1868, Marx delivers a lecture on the elements of
"the method of science" ', stressing among other things the importance
of the distinction between 'Erscheinung' and 'Wesen', 'appearance' and 'essence' in science.

In the "Preface" to the second edition
of Capital, Marx complains that "the method employed in "Das
Kapital" has been little understood", [6] and [1/2] takes even
pains to emphasize that "the method of presentation must differ in form
from that of inquiry". [7] Apparently it was no great help to the readers
of Capital that Marx assured them that his "dialectic method
is not only different from the Hegelian, but is its direct opposite". [8]
His German readers were too infatuated with Hegelianism or too little acquainted
with natural science and its methods to grasp the implication of the statement
that Hegel's dialectic "must be turned right side up again, if you would
discover the rational kernel within the mystical shell". [9]

In his attempts to come to grips with Marx's "dialectic
method" even Conrad Schmidt (1869‑1932), outstanding German economist
and Social‑Democrat, asked Engels for advice. Instead of referring him
to the model pattern of scientific method, rooted in natural science and employed
by Marx, Engels gave Schmidt the dubious advice, to "compare the development
of the commodity into capital in Marx with the development from Being <Sein>
to Essence <Wesen> in Hegel", assuring him that "you will get
quite a good parallel for the concrete development which results from facts".
[10]

Of course, it is true that some aspects of
Marx's method in social science are associated with Hegel's dialectic. [11]
However, the analysis of Marx's "dialectic method", as presented or
employed in the posthumous Introduction (1857), in Capital, in Marx's scientific letters, and in Mathematical Manuscripts reveals,
as a general characteristics, a dependence on classical physics and mathematics,
so far too little understood. The source of this dependence is, in the first
place, the scientific spirit of the "Scottish Historical School",
deeply rooted in Newtonian physical science, and, second, Marx's first‑hand
studies of Newton's mathematics and physics. [12]

Already in his Treatise of Human Nature (1739),
David Hume makes "an attempt to introduce the experimental method of reasoning
into moral subjects." [13] His intention was undoubtedly to do for "moral
philosophy" what Newton had done for "natural philosophy". In
his Essays, moral, political and literary, he makes the observation:
"So great is the force of laws, and of particular forms of government,
and so little dependence have they on the humours and tempers of men, that consequences
almost as general and certain may sometimes be deduced from them, as any which
the mathematical sciences afford us". [14] Therefore, "records of
wars, intrigues, factions, and revolutions, are so many collections of experiments,
by which the political or moral philosopher fixes the principles of his science;
in the same manner as the physician or natural philosopher becomes acquainted
with the nature of plants, minerals, and other external objects, by the experiments,
which he forms concerning them" [15].

Marx, who was familiar with Hume's discourses on
commerce, money, interest, balance of trade, etc. [16], was hardly impressed
by his attempts to apply the Newtonian method in social science. In his critique
of Hume's discourses [2/3] in A Contribution to the Critique of Political
Economy and Theories on Surplus Value [17] there is no direct
or indirect reference to or evaluation of Hume's “experimental method of
reasoning".

In Essays on Philosophical Subjects, "by the late Adam Smith",'" written prior to 1758,
we discover a treatise, "The Principles which lead and direct Philosophical
Enquiries; illustrated by the History of Astronomy" [19]. In his "Account
of the life and writings of Dr. Smith", the Scottish philosopher and economist,
Dugald Stewart (1753‑1828) writes that "the mathematical sciences,
both pure and mixed, afford, in many of their branches, very favourable subjects
for theoretical history; and a very competent judge, the late M. d'Alembert, has recommended this arrangement of their elementary principles, which is
founded on the natural succession of inventions and discoveries, as the best
adapted for interesting the curiosity and exercising the genius of students.
The same author points out as a model a passage in Montucla's History of
Mathematics [20], where an attempt is made to exhibit the gradual progress
of philosophical speculation, from the first conclusions suggested by a general
survey of the heavens, to the doctrines of Copernicus. It is somewhat
remarkable, that a theoretical history of this very science (in which we have,
perhaps, a better opportunity than in any other instance whatever, of comparing
the natural advances of the mind with the actual succession of hypothetical
systems) was one of Mr. Smith's earliest compositions, and is one of
the very small number of his manuscripts which he did not destroy before his
death" [21].

As indicated by its title, Smith's essay is not
a common history of astronomy. It is a treatise of method, containing the principles
that lead and guide scientific research in general. Of course, in
Smith such principles are those underlying Copernican‑Newtonian science.
Accordingly, it is the task of scientific research "to find out those hidden
chains of events which bind together the seemingly disjointed appearances of
nature" [22]. Since Sir Isaac Newton succeeded in doing this in a way without
precedence, "his system. . . now prevails over all opposition, and has
advanced to the acquisition of the most universal empire that was ever established
in philosophy. His principles, it must be acknowledged, have a degree of firmness
and solidity that we should in vain look for in any other system. The most sceptical
cannot avoid feeling this" [23].

In "Note by the Editor" we read: "The
Author, at the end of this Essay, left some Notes and Memorandums, from which
it appears that he considered this last part of his History of Astronomy as
imperfect, and needing several additions. The Editors, however, chose rather
to publish than to suppress it. It must be viewed, not as a History or Account
of Sir Isaac Newton's Astronomy, but chiefly as an additional illustration of
those Principles in the Human Mind which Mr. Smith has pointed out to be
the universal motives of Philosophical Researches" (italics mine) [24].
[3/4]

Although Smith employed the principles presented
in his Essays on Philosophical Subjects as a guide to scientific research
in the field of morals, sociology and economics, there are only vague allusions
to these principles in The Theory of Moral Sentiments [25] and The
Wealth of Nations [26]. As far as Marx is concerned, there is no textual
evidence in favour of the assumption that he knew the Essays and used
its principles as a guide to his studies. On the other hand, the reader
of Capital cannot help but observe that definite categories, assumptions
or procedures known and employed in natural science since the time of Copernicus
are consciously employed by Marx, and with great skill at that.

Again and again the reader of Capital comes
across the categories appearance <Erscheinung> and essence <Wesen>.
These terms are often called "dialectical" because they are used
by Hegel in Wissenschaft der Logik [27]. It is even maintained by Georg
Lukács in Geschichte und Klassenbewusstsein [28] and by Roman Rosdolsky
in Zur Entstehungsgeschichte des Marxschen "Kapital" [29] that
the Marxian distinction between appearance and essence stems from
Hegel. So far as Marx makes use of the terms in scientific texts, it does not.
It stems from the founders of modern natural science, and is rather a Platonic
feature of modern mathematical astronomy, and of classical mechanics in general.
The distinction seems to be connected with the Platonic requirement of "saving
appearances" through the discovery of the "real" or "essential"
movements of celestial bodies. This requirement played, as is well known, a
great role in ancient mathematicians and astronomers in their attempts to come
to grips with the complexity of astronomical phenomena, which presented itself
as "irregularities" or "deviations" [from the circular paths],
observed in the sun, the moon and, in particular, the planets [30].

It is easy to show that Marx employed the distinction
between essence and appearance in order to show how science, in contradistinction to pseudo‑science works. As examples of
science Marx quotes Copernican astronomy, anti‑phlogistic chemistry, and
classical economy, whereas pre‑Copernican astronomy, phlogiston theory
and vulgar economy serve as examples of pseudo‑science.

For instance, dealing with revenues and their sources
he says: "It should not astonish us, . . . that vulgar economy feels
particularly at home in the estranged outward appearances of economic
relations in which these prima facie absurd and perfect contradictions
appear and that these relations seem the more self‑evident the more their
internal relationships are concealed from it, although they are understandable
to the popular mind. But all science would be superfluous if the outward
appearance and the essence of things directly coincided" (italics
mine) [31].

The relationship between classical economy and vulgar
economy is accordingly described in the following way: "Once and for all
I may here state, [4/5] that by classical economy, I understand that
economy which, since the time of W. Petty (1623‑1687), has investigated
the real relations of production in bourgeois society, in contradistinction
to vulgar economy, which deals with appearances only, ruminates
without ceasing on the materials long since provided by scientific economy,
and there seeks plausible explanations of the most obtrusive phenomena, for
bourgeois daily use, but for the rest, confines itself to systematizing in a
pedantic way, and proclaiming for everlasting truths, the trite ideas held by
the selfcomplacent bourgeoisie with regard to their own world, to them the best
of all possible worlds" (italics mine) [32].

Marx never quotes passages out of Copernicus, although
he very often refers to the Copernican view, stressing the distinction between
appearance and essence. However, in order to give some textual evidence in favour
of the point of view that the Marxian distinction. between appearance and essence
originates in the Platonic-Copernican tradition, and not in Hegel, it may not
be amiss to quote Copernicus here.

In his Preface to De revolutionibus orbium
coelestium, addressed to Pope Paul II, Copernicus says that his friends
"insisted that, though my theory of the Earth's movement might at first
seem strange, yet it would appear admirable and acceptable when the publication
of my elucidatory comments should dispel the mists of paradox". . . "How
I came to dare to conceive such motion of the Earth contrary to the received
opinion of the Mathematicians and indeed contrary to the impressions of the
senses, is what your Holiness will rather expect to hear. So I should like
your Holiness to know that I was induced to think of a method of computing the
motions of the spheres by nothing else than the knowledge that the Mathematicians
are inconsistent in these investigations". . . Therefore, "those
who have relied in concentrics, though they have proven that some different
motions can be computed therefrom, have not thereby been able fully to establish
a system that agrees with the phenomena" [33]; that is, they have not been
able to "save phenomena".

As stated, Copernicus had one fundamental principle
in common with his immediate predecessors in astronomy. It was formulated as
the requirement that the so‑called "irregularities" or "deviations"
of celestial bodies should be "reduced" to circular movements. In
another respect he heartily disagreed with his predecessors because he arrived
at the insight that their solution of the problem was inconsistent. This inconsistency,
he argued, had its roots in the immediate testimony of the senses, informing
you that the Earth is at rest.

The next step was: choice of a different frame of
reference, the sun; analysis of all data so far accumulated, including own observations,
and combination of data to a consistent whole. The outcome of these endeavours
was the propositions contained in the Copernican theory of the movement of celestial
[5/6] bodies, at the basis of which lay the assumption that the Earth
has a threefold movement. This theory, that supposedly described the "real"
movements of celestial bodies, made understandable the immediate testimony
of the senses and explained the "apparent" movements of celestial
bodies.

As we shall see, Marx used precisely the same
model‑pattern of explanation in social science.

The Copernican way of talking about science and
scientific method in terms of appearance and essence is partly out of date.
However, as we perceive, it plays a fundamental role in Marx, and this to such
extent that Capital hardly is understandable, if this mode of speaking
is not accepted.

In Wages, Price and Profit, written
1865, we discover the following passage: "Of course, having once found
out the true but hidden sense of the expression 'Value of Labour', we shall be able to interpret this irrational, and seemingly impossible
application of value, in the same way that, having once made sure of the real
movement of celestial bodies, we shall be able to explain their apparent or
merely phenomenal movement" [34].

In Capital Marx continues the same trend
of thought, saying that "in the expression 'value of labour', the idea
of value is not only completely obliterated, but actually reversed. It is an
expression as imaginary as the value of the earth. These imaginary expressions,
arise, however, from the relations of production themselves. They are categories
for the phenomenal forms <Erscheinungsformen> of essential <wesentlicher>
relations. That in their appearance things often represent themselves in inverted
form is pretty well known in every science except political science" [35].

Referring to "every science except political
science", Marx has hardly Hegel's Wissenschaft der Logik in mind,
but rather Copernican astronomy, Newtonian mechanics, and anti‑phlogistic
chemistry. This point of view is confirmed by the following passage: ".
. .this much is clear; a scientific analysis of competition is not possible,
before we have a conception of the inner nature of capital, just as the apparent
motions of the heavenly bodies are not intelligible to any but him, who is acquainted
with their real motions, motions which are not directly perceptible by the senses"
[36].

Apart from the basic distinction between appearance
and essence, Marx takes over from physical science procedures that are only
partly described or applied by Copernicus. In Newton they are in full bloom.
We have already mentioned the distinction between the method of inquiry and
the method of presentation, and may add that the former includes
analysis of conceptual systems, developed by predecessors in the field of investigation;
conceptual analysis. and constructive activity as part of thought experiments;
indirect proof or reductio ad absurdum, involving consistency
requirement; sifting and analysis of data; abstraction, idealization, and "mathematical"
generalization, including the creation of new concepts and formulation of new
hypotheses; [6/7] application of the criterion of recurrence, with the
discovery of "laws of motion" in view; formulation of such laws in
mathematical terms. The method of presentation, on the other hand, includes
the creation of an adequate system of symbols, and the construction of a deductive
chain, connecting the elements of the structure.

As stated above, several contemporary readers of
A Contribution to the Critique of Political Economy and of Capital
were so unfamiliar with scientific method that they failed to grasp Marx's
scientific method and achievements. Consequently, Marx felt obliged to make
several notes on the problem of reading and understanding Capital. And he did so in order to stand up against some misunderstandings, amounting
to the accusation that the theories presented in Capital were a priori
constructions. On this occasion he wrote in the Preface to the second
edition of Capital: "Of course the method of presentation
must differ in form from that of inquiry. The latter has to appropriate the
material in detail, to analyze its different forms of development, to trace
out their inner connection. Only after this work is done, can the actual movement
be adequately described. If this is done successfully, if the life of the subject‑matter
is ideally reflected as in a mirror, then it may appear as if we had before
us a mere a priori construction" [37].

However, to a reader familiar with Newton's Principia,
this is a matter of course. A hundred years ahead of Marx, the author of An
Essay ofthe History of Civil Society [38], Adam Ferguson,
Smith's teacher, published a textbook for students in the College of Edinburgh,
Institutes of Moral Philosophy, where he gives a summary of the
method which he recommends for the study of moral and social phenomena. Here
we discover the following passage: "Method in science is of two kinds;
analytic and synthetic. Analytic method is that by which we proceed from observation
of fact, to establish general rules. Synthetic method, is that by which we proceed
from general rules to their particular applications. The first is the method
of investigation. The second of communication, or of the enlargement of science”
[39]. Or, as Marx put it, "the method of presentation", "Argument",
Ferguson continues, "is of two kinds: A priore and a posteriore.
By an argument a priore, the fact is proved from the law. By an argument
a posteriore, the law is proved from the fact" [40].

Accordingly, "theory consists in referring
particular operations to the principles, or general laws, under which they are
comprehended; or in referring particular effects to the causes from which they
proceed. To point out any general rule of law of nature previously known, in
which any particular fact is comprehended, is to account for the fact. Thus
Sir Isaac Newton accounted for the planetary revolutions, by showing that they
were comprehended in the laws of motion and gravitation" [41].

It is here interesting to compare these passages
to Newton's “Preface" to the first edition of Principia, where lie
points out that "the whole burden [7/8] of philosophy seems
to consist in thisfrom the phenomena of motions to investigate the forces
of nature, and then from these forces to demonstrate the other phenomena"
[42]. As far as the method of inquiry and the method of presentation are concerned,
such comparison seems to establish a link between the founder of classical mechanics
and the founder of classical sociology.

If we cast a glance at Marx's method of investigation,
we discover a set of elements, well­known from physical science. In the first
place, it involves analysis of conceptual systems, handed down to posterity
by predecessors in the field. The list of predecessors enumerated by Marx comprizes
more than 300 authors and almost as many conceptual systems. The analysis of
such conceptual systems, dealing with philosophical, social, and economic problems
is called "critique". Such critique is discovered in A Contribution
to the Critique of Hegel's Philosophy of Law, in The German Ideology, with the subheading Critique of Recent German Philosophy in its Representatives
Feuerbach, B. Bauer and Stirner, in Economic‑Philosophical
Manuscripts of 1844, and of course in AContribution to the Critique
of Political Economy, in Capital, and in Theories
of Surplus‑Value.

Secondly, Marx's method of investigation comprizes sifting
and analysis of historical, empirical and statistical material, provided by
historical sources, Parliamentary Papers and Reports, Agricultural Statistics,
Reports from Embassies, Children's Employment Commission Reports, Public Health
Reports, Factory Inquiry Commission Reports, Reports of the inspectors of factories,
Statistical Abstract for the United Kingdom, and a long series of Periodica
from Bengal Hurkaru til Vestnik Evropy and the Workman's Advocate.

Although Marx's subject‑matter of study and,
consequently, the material collected for analysis, sifting, categorization and
classification obviously is basically different from that of the physicist,
Marx's aim and procedures, as far as a project of research is at stake,
is analogous to that of the physicist. In the "Preface" to
the first edition of Capital we read that "it is the ultimate aim
of this work, to lay bare the economic law of motion of modern society"
[43] or "the natural laws of capitalist production" [44]. Already
in the "Preface" to A Contribution to the Critique of Political
Economy we read that "the economic conditions of production. . . can
be determined with the precision of natural science” [45].

The method of investigation includes, consequently,
an analysis of conceptual systems, guided by the requirement of consistency;
and, concomitantly, sifting, analysis, categorization, and classification of
historical, empirical and statistical material. This is done with a view to
the discovery of the "essence" or "inner connection""
of definite social phenomena, and to a subsequent formulation of laws of motion
that enable the investigator to account for the social phenomena investigated.
[8/9]

In a letter from April 30, 1868, Marx gives
a short sketch of his method of presentation, including a plan of presentation
of the subject of vol. Il and III of Capital. The reader (Engels)
is also made acquainted with a system of symbols, representing economic magnitudes.
As one might expect the subject is presented in terms of "essence"
and "forms of appearance", except that the term "essence"
is replaced by the term "law".

Marx gives the following summary: "The laws.
. . discovered will be, e.g., very important for understanding how the price
of raw material influences the rate of profit and they hold good no matter
how the surplus value may be later divided between the producer, etc."
. . . "At last we have arrived at the forms of appearance which
serve as the starting point in the vulgar conception: ground rent coming
from the earth, profit (interest) from capital, wages from labour. But from
our point of view the thing is now seen differently. The apparent movement
is explained (italics mine). Moreover, A. Smith's nonsense, which has become
the main pillar of all economics hitherto, that the price of a commodity
is derived from those three revenues, i.e., only from variable capital (wages)
and surplus value (ground rent, profit, interest), is overthrown" [47]".

In the "Preface" to the first edition
of Capital Marx takes pains to point out the difference between
the procedures employed by the physicist, chemist or biologist, and his own
procedure. "In the analysis of economic forms. . . neither microscope nor
chemical reagents are of use. The force of abstraction must replace both"
[48]. That is, in sociology and economics you cannot make observations in connection
with an experimental set‑up. On the other hand, however, Marx's method
of abstraction is analogous to the method of abstraction and idealization in
physics. This method, moreover, involves a thought experimental procedure, structurally
similar to procedures known from the history of classical and modern physics.

Preliminarily, we may say that "the force of
abstraction" resides in the device of "leaving aside", "disregarding"
or "thinking away" definite aspects of the objects investigated. What
we "abstract from", "leave aside" or "put outside brackets"
is usually called "accidental" or "contingent" aspects of
the objects. In spite of possible objections to this modus loquendi we
adhere to it, not only because it is used by Marx [49], but also because it
is employed by classical, and even modern physicists.

When abstraction is defined in such terms, it is
almost obligatory to quote Galileo's Dialogues Concerning Two New Sciences,
and we do so in order to show, what it involves to renounce "real"
experiments, and to rely solely on "the force of abstraction". In
Dialogues we read: "We have. . . seen that the difference of speed
between bodies of different specific gravities is most marked in those media
which are the most resistant: thus, in a medium of quicksilver, gold not merely
sinks to the bottom more rapidly than lead but it is the only substance that
will descend at all; all other metals and stones rise to the surface [9/10]
and float. On the other hand, the variation of speed in air between balls
of gold, lead, copper, porphyry, and other heavy materials is so slight that
in a fall of 100 cubits a ball of gold would surely not outstrip one of copper
by as much as four fingers. Having observed this I came to the conclusion that
in a medium totally devoid of resistance all bodies would fall with the same
speed" [50].

To this way of reasoning, a modem physicist, Gerald Holton,
has the following comment: "As Galileo reveals his attitude toward experiments,
we are reminded of the old quip that 'science has grown almost more by what
it has learned to ignore than by what it has had to take into account'. Here
again we meet the recurring theme in science: Regard observable events with
a penetrating eye to search behind the immediate confusion of appearances for
an underlying simplicity and mathematical lawfulness. . . To arrive at the correct
result, it would have been futile to rely on the observational methods available
at the time; instead, everything depended on being able to 'think away' the
air and its effect on free fall. This is easy enough for us who know of pumps
and the properties of gases; but it was at that time conceptually almost impossible
for several reasonssome evident, but some quite subtle. For example,
Aristotle, who had examined the question of the existence of a vacuum, had denied
its possibility as contrary not only to experience but to reason. One of his
arguments was: Where there is a vacuum, there is nothing. If nothing remains,
he continued, then also space itself and the physical laws can no longer exist
in this vacuum; for how could a stone, for example, know which way is up or
down except in relation to and through the influence of a medium? The answer
depended on dissociating the words vacuum and nothingness, and so preventing
space from becoming annihilated merely because air has been withdrawn. But though
Galileo found it possible to do this in thought, Descartes and his followers
continued to reject the possible existence of a vacuum, and we may believe that
other scientists. eventually accepted it mainly on the basis of subsequent experimental
demonstrations that bodies could fall, and that heat, light, magnetism, etc.,
could be propagated through vessels from which air had been withdrawn"
[51].

In order to grasp the impact of classical physics
on Marxian social science, it may not be superfluous to call to mind this elementary
pattern, and point out that the steps taken by the founders of classical mechanics
from celestial to terrestrial phenomena were steps into greater and greater
complexity. Already freely falling bodies raised problems that required extraordinary
"abstractive power" and far‑going idealization at that. It was
the beginning of the time, when bodies were supposed to fall in vacuum, although
no vacuum could be produced; where ideal gases, perfectly elastic bodies, mass‑points,
inertial systems were "constructed", and where, for instance, a pendulum
was transformed into a mysterious heavy body without extension, that oscillated
[10/11] in a weightless cord without friction. It became more and more
common from the time of Simon Stevin and Galileo to Christian Huygens and Isaac
Newton to investigate bodies, divested of their "natural" attributes,
or equipped with more or less phantastic properties, for the purpose
of discovering the laws governing change or motion of physical bodies and their
relationships. Marx was keenly aware of this aspect of scientific work.

Textual evidence of this view is discovered in the
posthumous "Introduction" (1857) to A Contribution to the Critique
of Political Economy (1859), and throughout Capital. Here we come
across several passages where "the force of abstraction" is combined
with the procedure of idealization and employed in the search for "the
inner organization" or "the law of motion" of the society investigated.
For instance, "in our description of how production relations are converted
into entities and rendered independent in relation to the agents of production,
we leave aside the manner in which the interrelations due to the world‑market,
its conjunctures, movements of the market-­prices, periods of credit, industrial
and commercial cycles, alterations of prosperity and crisis, appear to
them as overwhelming natural laws that irresistibly enforce their will over
them, and confront them as blind necessity. We leave this aside because the
actual movement of competition belongs beyond our scope, and we need only the
inner organization of the capitalist mode of production in its ideal average,
as it were" (italics mine) [52].

In the "Preface" to the first edition
of Capital, Marx informs the reader that he has "to examine the
capitalist mode of production, and the relations of production and exchange
corresponding to that mode" [53]. However, as we learn from the "Introduction"
(1857), the starting‑point of his investigation is "a chaotic representation
or conception of the whole" [54], namely contemporary British society.
This is where the method of inquiry goes into action.

In the first place, almost all conditions for and
aspects of life in this society are taken into consideration: the geological,
geographical and climatic conditions; the population; procreative, material
and intellectual production, and the corresponding relations of production;
division of labour: agriculture. handicraft, industry, trade, intellectual activity
in art, philosophy, science, politics, civil service; classes: peasants, artisans,
and wage‑labourers, employed and non‑employed, workers and capitalists:
industrial capitalists, merchants, money‑lenders, landowners; intellectuals
and civil servants: artists, philosophers, scientists, economists, doctors,
inspectors, directors, churchmen, journeymen, politicians, military persons,
police etc. etc.

In short, the investigator is confronted with a
collection of facts, events. activities and processes, more or less adequately
described or reported ill historical, 'economical and political works; in art,
religion, philosophy: in natural and social science, and, last not least, in
social statistics. Confronted with such complexity, the main problem is: how
can research have access [11/12] to this chaos? How does one arrive at
the core of the problem posited? How can the criterion of recurrence, known
from natural science possibly be employed to social phenomena?

It is well‑known that Marx, in his attempt
to employ this criterion, was guided by definite ontological assumptions, inherent
in his philosophical materialism and anthropology. Here it may be sufficient
to refer to The German Ideology [55], and to the "15 sentences"
from the "Preface" to A Contribution to the Critique of Political
Economy [56]. It should be noted, however, that in order to be able
to employ the criterion of recurrence, it was necessary, preliminarily, not
only to "leave aside" or "put outside brackets" development
proper, that is, the central subject of dialectics, which Marx,
primarily, illustrated by the transition from one social formation or system
into another; it was also necessary to take the following three decisive abstractive
steps.

According to basic assumptions, underlying Marxian
social science, one cannot account for social phenomena with reference to physical
or biological phenomena; consequently, social phenomena cannot be described
in terms of physics or biology. In modern parlance: Marx was no "reductionist".
Hence, in attempts to consider a given country from a politico‑economic
standpoint, one should not start inquiry with analysis of physical or biological
"forces" that originate in the geological, geographical or climatic
environment. The specific or variable character of the environmentwhich
today has come into focus in ecological researchis "left aside"
or "disregarded"; although definite elements, present in any environment
where human beings live and "produce their means of subsistence",
are taken into account in one of the basic concepts of Marxian social science:
the concept of means of production.

In the "Introduction" (1857) we discover
the following remark: "It would seem to be the proper thing to start with
the real and concrete elements, with the actual pre‑conditions, e.g.,
to start in the sphere of economy with population, which forms the basis and
the subject of the whole social process of production" [57]. This turns
out to be wrong, however, because one cannot account for social phenomena in
terms of race, temperament, national character, growth, decrease, mobility of
the population. These aspects are consequently considered "accidental",
from a politico‑economic point of view. Although these aspects are "left
aside" at this initial stage of inquiry, the population is not "forgotten".
It is taken into account in several basic concepts of Marxian social science:
the concept of productive forces, the concept of relations of production, and the concept of classes.

In the enumeration of facts or phenomena pertaining
to human society, and to capitalist society in particular, wants, needs or drives
were not mentioned, although they are introduced on the first page of chapter
one in A Contribution to the Critique of Political Economy and in Capital,
in con[12/13]nection with a preliminary definition of commodity. "A commodity is, in the first place, an object outside us, a thing
that by its properties satisfies human wants <Bedürfnisse> of some sort
or another. The nature of such wants, whether for instance they spring from
the stomach or from fancy, makes no difference" [58].

As the student of Capital is well
aware, wants or needs play a very secondary role in Marx's works, as do use‑values
in contradistinction to values. Their relationship is brought
out in the remark: "To say that a commodity has a use‑value is merely
to say that it satisfies some social want" [59]. We may even say that,
in Marx's presentation of the structure of capitalist society, wants or needs
are "left aside" or "thought away".

This abstractive step is founded on the conception
that the subject matter of social science is, primarily, the relations between
man and man, and only in a derivative manner, the relations between man and
nature. This involves that, in the search for social phenomena on which the
criterion of recurrence can be applied, productive activity and the corresponding
social relations of production and exchange, related to values, are brought into focus; whereas consumption, related to wants, needs or
use­-values, is put into the background at this initial stage
of inquiry.

In conformity with these abstractive steps, i.e.,
according as environment, population, needs, wants, and drives are "left
aside" or "thought away" as specified, and taken into consideration
only in as far as they are the basis of the entire act of production,
a totality of social activities and relations are left for analysis. In Marx
the basic relations are those connected with procreative production or production
of human beings, "material" production or production of "material"
goods, and intellectual <geistige> production or production of language,
art, religion, philosophy, science, morals, law, ideology. etc.

The subsequent abstractive step involves a choice, based
upon ontological assumptions inherent in Marxian philosophical materialism and
anthropology. Of special importance is the assumption that "men. . . themselves
begin to distinguish themselves from animals as soon as they begin to produce
their means of subsistence" [60], and begin to establish the corresponding
specific human relations of production. Guided by these assumptions Marx
decides that the criterion of recurrence cannot possibly be applied to the relations
of production connected with intellectual production. These social relations
are therefore "left aside" for the time being. As far as the relations
of procreation are concerned, they are "left aside" in favour of the
specific human social relations that are known, plainly, as "relations
of production".

Arrived at this stage, another set of abstractive
steps are taken. For with the understanding that we are investigating capitalist
Great Britain in the [13/14] middle of the 19th century, we are faced
with relations of production and exchange of a definite type, that is,
with social relations which appear as capital. Here, in modem bourgeois
society, Marx discovers different forms of capital: merchants' capital, interest‑bearing
(money‑lenders') capital, and, last not least, industrial capital. Hereto
is added landed property.

Before analysis proceeds, two questions must be
answered: how are the relations of production in capitalist society connected
with capital? Which are the relationships between the different forms of capital?
The first question is answered in the following remark: "Capital is not
a thing, but rather a definite social production relation, belonging to a definite
historical formation of society, which is manifested in a thing and lends this
thing a specific social character. Capital is not the sum means of production
transformed into capital, which in themselves are no more capital than gold
and silver in itself is money” [61].

Since the question concerning the relationships
between the different forms of capital is a question concerning the order of
logical precedence, analysis of capitalism cannot start before the proper answer
to this question is furnished. Now, landed property existed in Antiquity, and
"nothing seems more natural than to begin with rent i.e., with landed property,
since it is associated with the earth, the source of all production and all
life, and with agriculture, the first form of production in all societies that
have attained a measure of stability". Hence, why not start analysis with
landed property?

The answer is: "Nothing would be more erroneous"
[62]. This is so because "there is in every social formation a particular
branch of production which determines the position and importance of all the
others, and the relations obtaining in this branch accordingly assign rank and
influence to all other branches" [63]. In capitalist society, "agriculture
to an increasing extent becomes just a branch of industry and is completely
dominated by capital". "The same applies to rent". "Capital
is the economic power that dominates everything in bourgeois society".
Consequently, "rent cannot be understood without capital, but capital can
be understood without rent" [64].

Merchants' and money‑lenders' capital are,
historically, ahead of industrial capital. It would therefore seem natural to
start analysis of capitalism with these forms of capital. There are, however,
serious obstacles to such procedure. For one thing, one would be justified in
maintaining the absurdity that capitalism came into being already in ancient
Athens or Rome. This was, in fact, what Karl Kautsky did in his Der Ursprung
des Christentums [65]. Secondly, attempts to define capitalism in terms
of merchants' capital or in terms of money‑lenders' capital would clash
with Marx's method of inquiry, in particular with the requirement concerning
the order of logical precedence.

Marx argues as follows: "In the course of our
investigation, we shall find that both merchants' capital and interest‑bearing
capital are derivative forms, [14/15] and at the same time it
will become clear, why these two forms appear in the Course of history before
the modern standard form of capital" [66] (italics mine).
Consequently, the standard form of capital, i.e., industrial capital, is singled
out as the defining mark of capitalism. Industrial capital, again, is defined
in terms of a definite social relation, viz. the relation between the industrial
capitalist and the wage‑worker. Hence, "in analyzing the standard
form of capital, the form under which it determines the economical organization
of modern society, we entirely left out of consideration its most popular,
and, so to say, antediluvian forms, merchants' capital and money‑lenders'
capital"" (italics mine).

With these abstractive steps we have, preliminarily,
reached the pinnacle of abstraction. In a step‑by‑step fashion all
physical and biological factors, irrelevant in the given context have been "thought
away"; and all social relations except the relation between the class of
industrial capitalists and the class of wage‑workers have been "left
out of consideration". Consequently, we have arrived at a rather odd capitalist
society, very far from "the chaotic conception of the whole", with
which investigation started. Our object of study has now been transformed into
a "capitalist mode of production in its ideal average”, into an ideal
capitalist society, to which the criterion of recurrence can be applied.
However, the employment of this criterion involves a set of procedures that
call to mind the thought‑experimental procedures that played such great
role in the development of classical physics.

If we turn the pages of the first volume of Capital, we perceive that chapters I‑III deal with commodities and money. Capital
is not introduced until chapter IV. Why is this so? Why does the exposition
of the subject not start with the presentation of industrial capital, which
is supposed to be the defining mark of capitalism?

By closer inspection of industrial capital, it turns
out that the social relation between the class of industrial capitalists and
the class of wage‑workers is a double relation, viz. a relation of production
and a relation of exchange. It is even, primarily, a relation of
exchange because the social relation between the industrial capitalist and the
wage­worker is established as an exchange relation: the worker offers
"something". for sale on the market which the industrial capitalist
buys; and it becomes a relation of production only in as far as the industrial
capitalist uses this "something" in an act of production. Now, analysis
reveals that this exchange relation belongs to the set of commodity exchange
relations; and in accordance with the analytical procedure suggested in the
"Introduction" (1857) [68], the starting point for the analysis of
this "real and concrete" relation should be the exchange relation
in its "more simple" form. This methodological device leads, of course,
to another abstractive step, i.e., to the investigation of exchange relations
under conditions where industrial capital does not exist, where producers (peasants
[15/16] artisans) own their means of production, and where these products
(are not consumed by the producers themselves, but) are taken to the market
for alienation as "use‑values for others" [69] or commodities
in the Marxian sense.

As one perceives, this is only possible in a commodity
producing society of a very elementary type. And the question naturally arises:
has such precapitalist commodity producing society ever existed in history?
Marx does not answer this question in a univocal way, although there are passages
in Marx which testify strongly in favour of an affirmative answer.

The following passage is illuminating: "The
private property of the labourer in his means of production is the foundation
of petty industry, whether agricultural, manufacturing or both; petty industry,
again, is an essential condition for the development of social production and
of the free individuality of the labourer himself. Of course, this petty mode
of production exists also under slavery, serfdom, and other states of dependence.
But it flourishes, it lets loose its whole energy, it attains its adequate classical
form, only where the labourer is the private owner of his own means of labour
set in action by himself: the peasant of the land which he cultivates, the artisan
of the tool which he handles as a virtuoso. This mode of production pre­supposes
parcelling of the soil, and scattering of the other means of production. . .
It is compatible only with a system of production, and a society, moving within
narrow and more or less primitive bounds. To perpetuate it would be, as Pecqueur
rightly says, "to decree universal mediocrity" [70]. Mostly Marx refers
to such mode of production in terms of "pre‑capitalist" or "simple
exchange of commodities" [71].

Whether or not we can accept the historical existence
[72] of a society of this type, we can hardly refuse to accept the assertion
that the exchange relation between the industrial capitalist and the wage‑worker
belongs to the set of commodity exchange relations; and, that it is recommendable
to analyze these exchange relations in their "simplest and most abstract
form", in order to have a basis for the analysis of the "more complex
or concrete forms". In fact, Marx constructs a highly idealized
society or a model‑society, whose members are small peasants and artisans
who own their means of production and products, and who alienate these products
on the market in exchange for other products. With this model‑society,
whose existence he postulates [73], Marx makes his first great thought
experiment. In particular, he employs the criterion of recurrence to definite
social relations connecting its members. For instance, it is pointed out that
"the constant repetition of exchange makes it a normal social act"
[74]', or that "the currency of money is the constant and monotonous repetition
of the same process" [75]. This feature of constancy or invariance is expressed
in its laws of production and exchange.

Since the theory of this model‑society, as
presented in chapters I‑III of Capital [76], contains precise definitions
of the concept of commodity, the con[16/17]cept of value, the concept
of exchange, and the concept of money, plus definite laws of production
and exchange, usually summarized as "the law of value", one may be
justified in calling this theoretical structure "the theory of pre‑capitalist
or simple commodity producing society". Here, for short, it is called
theory I. It was presented, for the first time in 1859 in A Contribution
to the Critique of Political Economy.

From a political point of view this book was a failure.
And no wonder! Although Part One (there was no Part Two) had the subheading,
"Capital in general'', it contained nothing about capital or capitalism,
not to speak of the transition from capitalism to socialism. Cp. Wilhelm Liebknecht's
and Elard Biskamp's commentaries! [77] The book ends with "simple commodity
circulation" [78], and stops in so far at the most exciting scene of action,
viz. where ''the transformation of money into capital" is on the agenda.

In a letter from January 13, 1859, Marx gives a
report on this anomalous situation, but does not explain it. It would seem,
however, that Marx at that time was confronted with a serious logical problem,
to which no satisfactory solution was provided until the middle of the sixties.
The problem concerned the concept of labour and the possible application of
the labour theory of value to capitalist society. As is well‑known, Ricardo
failed to solve this problem. A note on the situation is discovered in a letter
from June 24, 1865, where Marx informs Engels that he, in an address delivered
on June 20 and 27, 1865, at the two sittings of the General Council of the International
Working Men's Association, had "borrowed many new aspects" [79] from
the manuscript to Capital. Here, in Part II, "The transformation
of money into capital", the final solution of the problem is presented
for the first time.

In his address, June 1865, Marx points out that
the attempt to conceive labour as a commodity or to talk about "the value
of labour" leads to "a seeming paradox" or to "nonsensical
expressions" [80]. Since this was the way in which Ricardo talked about
the relation between the industrial capitalist and the labourer, there was certainly
a problem inherent in the labour theory of value.

Before it is shown, how Marx, through a rather primitive,
almost purely analytical or critical thought experiment, discovers an error
in Ricardo's reasoning, and demonstrates that his application of the laws of
production and exchange, according to the labour theory of value, leads to a
paradox, it may be useful to look at a classical thought experiment in physics,
that hardly is less primitive than that performed by Marx. It involves, as Kant
would say, solely an "analysis of the concepts which we already have of
objects" [81], and relies exclusively on "the highest principle of
all analytical judgments". viz. "the principle of contradiction"
[82]. Nevertheless, it changed basically our way of thinking about physics.
It caused a revolution in science.

In Dialogues Concerning Two New Sciences Galileo
investigates, from different angles, Aristotle's conception of freely failing
bodies. We have al-[17/18]ready mentioned one angle [83]. Here is another:
in Aristotle gravity is an essential property of physical bodies, and his argument
concerning freely falling bodies starts from the correct premiss that the gravitational
pull of a body weighing 10 pounds, say, is ten times stronger that the gravitational
pull of a body weighing 1 pound. From this premiss he makes the inference: two
stones, one weighing 10 times as much as the other, if allowed to fall, at the
same instant, from the height of, say, 100 cubits, would so differ in speed
that when the heavier had reached the ground, the other would not have fallen
more than 10 cubits.

Galileo, seeing the fallacy, argues that it is possible,
even without further experiments, to prove clearly, by means of
a short and conclusive argument, that a heavier body does not move more rapidly
that a lighter one provided both bodies are such as those mentioned by Aristotle.
"If then we take two bodies whose natural speeds are different, it is clear
that on uniting the two, the more rapid one will be retarded by the slower,
and the slower will be somewhat hastened by the swifter. . . But if this is
true, and if a large stone moves with a speed of, say, eight while a smaller
moves with a speed of four, then when they are united, the system will move
with a speed less than eight; but the two stones when tied together make a stone
larger than that which before moved with a speed of eight. Hence the heavier
body moves with less speed than the lighter; an effect which is contrary to
your <i.e., Aristotle's> supposition" [84].

However, the discovery of this paradox did not imply
a disclosure of the root of the Aristotelian fallacy. It only paved the way
for the introduction of the concept of inertial mass, which Aristotle ignored
and consequently failed to take into account in his argument. This discovery
was reserved to Newton, who did not only introduce the explicit distinction
between inertial and gravitational mass, but also furnished the proof that they
are proportional. Consequently he was able to explain that "all
sorts of heavy bodies descend to the earth from equal heights in equal
times" [85]. It is noteworthy that Newton arrived at this conclusion in
a series of analytical and constructive thought experiments, which served as
a model‑pattern of research for generations of scientists. Adam Smith
[86] was one of them.

If we compare Adam Smith's and Hegel's assessment
of Newton's method in science, the discrepancy between their evaluations is
so striking that it seems worthwhile to ask the question: did Marx really, in
conformity with Engels, accept Hegel's depreciatory evaluation of Newton's conceptual
system and procedures in science, or was he rather guided by the view of the
matter adopted in "The Scottish Historical School"? The question is
important in several respects. For one thing, because Engels in Dialectics
of Nature, on the whole, approved of the verdict passed by Hegel
upon Newtonian physics, and even went so far as to call Newton an "Induktionsesel"
[87]. Secondly, because the role of thought experiments in science was the point
at issue. [18/19]

In his History of Astronomy Adam Smith
gives a lucid and correct description of Newton's thought experiment with "the
Moon . . . conceived as constantly failing towards the Earth" [88]. This
crucial thought experiment was prepared in a series of thought experiments,
starting with reflections concerning the relationship between courses taken
by bodies released on the top of a mountain, moving uniformly in a right line
parallel to the horizon with different given velocities, starting from, say,
zero: "If a leaden ball, projected from the top of a mountain by the force
of gunpowder, with a given velocity, and in a direction parallel to the horizon,
is carried in a curved line to the distance of two miles before it falls to
the ground; the same, if the resistance of the air were taken away, with the
double or decuple velocity, would fly twice or ten times as far. And by increasing
the velocity, we may at pleasure increase the distance to which it might be
projected, and diminish the curvature of the line which it might describe, till
at last it should fall at the distance of 10, 30, or 90 degrees, or even might
go round the whole earth before it falls; or lastly, so that it might never
fall to the earth, but go forwards into the celestial spaces, and proceed in
its motion in infinitum" [89].

In his Naturphilosophie, Hegel has
the following comments to these reflections: "The projection exhibits the
accidental movement in contradistinction to the essential <movement> of the fall; however, the abstraction, the body as it body, is inextricably
connected with its gravity: and, in the projection this gravity, consequently,
thrusts itself, by itself, into being taken into consideration. The projection
as isolated, existing separately, cannot be shown. The example
of the movement, supposed to originate in vis centrifuga, is usually
the stone in it sling, that, whirled around in an orbit by the hand, always
shows the tendency to recede from it" [90].

The continuation of this text is quoted by Engels
in notes written towards the end of 1877 and beginning of 1878, after the publication
in separate form of the first section (Philosophy) of "Anti‑Dühring"
[91]: "But the point is not that such tendency exists, but
that it exists for itself separate from gravitation, as conceived
in a completely independent form in force. In the same place,
Newton assures us that a lead bullet in coelos abiret et motu abeundi pergeret
in infinitum <might go forward into the celestial spaces and proceed
in its motion in infinitum>, if (certainly if) only the
appropriate velocity Could be imparted to it. Such separation of external
and essential motion belongs neither to experience nor to the notion but only
to abstracting reflection. It is one thing to distinguish them, as is
necessary, as well as to characterize them mathematically as separate lines,
to treat them as separate quantitative factors, and so on it is
another thing to regard them as physically independent existences" [92]
.

These passages, and Hegel's Naturphilosophie
in general. bear witness to a quasi­Aristotelian conception of the physical
world, in which the legitimacy [19/20] of basic concepts and analytical
devices, developed and employed by the founders of classical mechanics is challenged.
It may here be sufficient to mention the following items:

Hegel does not see the point in defining force in
terms of acceleration and inertial mass. Nor does he grasp the significance
of introducing the concept of inertial mass in order to make the behaviour
of freely falling bodies understandable. Consequently, he ignores the proof
of the proportionality of inertial and gravitational mass [93]. Since he rejects
the role of abstraction, idealization and the thought experimental procedure,
he refuses to accept Newton's first law of motion, and rejects the legitimacy
of constructing a "mathematical pendulum" [94] for the purpose of
establishing the relationship between the length of a pendulum and its oscillatory
cycle. Moreover, as quoted above, he repudiates the possibility of a "gravity‑free"
space, much in the same way as Aristotle denies the possibility of "empty"
space. Finally, Hegel does not accept the thought experiment with the moon that
served as a basis for the proof of the proportionality of inertial and gravitational
mass, and for the final discovery and formulation of the law of gravitation
[95]. For the sake of argument, it may be useful to quote a few passages from
Newton's report on this experiment: "And now if we imagine the moon
deprived of all motion, to be let go, so as to descend towards the earth
with the impulse of all that force by which (by Cor. in Prop. III) it is retained
in its orb, it will in the space of one minute of time, describe in its fall
15 1/12 Paris feet . . . Wherefore, since that force, in approaching to the
earth, increases in the proportion of the inverse square of the distance, and,
upon that account, at the surface of the earth, is 60.60 times greater than
at the moon, a body in our regions, failing with that force, ought in the space
of one minute of time, to describe 60.60.15 1/12 Paris feet; and, in the space
of one second of time, to describe 15 1/12 of those feet as Huygens has observed
in experiments with pendulums. "And therefore (by Rule I and 2) the force
by which the moon is retained in its orbit is that very same force which we
commonly call gravity". . . "This calculus is founded on the hypothesis
of the earth's standing still" (italics mine) [96].

As made plausible above, Capital and its
immediate forerunners give ample evidence in favour of the assumption that Marx
was familiar with the analytical devices and methodological procedures developed
by the founders of classical mechanics, and that he was able to employ them
with great skill. He hardly shared Hegel's twisted view of Newtonian mechanics,
and could not possibly sanction his rejection of the method of abstraction and
idealization or his denial of the legitimacy of the thought experimental procedure.
Marx's methodological approach to social phenomena had, indeed, very little
in common with Hegel's so‑called "dialectical method". It was
rather related to the method adopted in "The Scottish Historical School",
[20/21] refined by Marx's own creative appropriation of the scientific
spirit inherent in classical mathematics and natural science.

It would therefore seem that the above digression
from Marx's method of investigation and the trip into the field of mechanics
might be helpful for the understanding of his employment of the thought experimental
method, and other analytical and constructive devices.

Classical economists, Ricardo in particular, was
faced with the problem: what is the nature of the relation of exchange between
the labourer and the industrial capitalist? What is the "something"
that the labourer alienates as it commodity, and how is this "something"
used by the industrial capitalist? Ricardo identifies this "something"
with labour. It was therefore natural to apply "the law of value",
according to the labour theory of value, to the analysis of the relation of
exchange and production between labourer and capitalist.

Ricardo argued, according to Marx, in the following
way: 1. labour is the "source" of value. Hence, according to the law
of value, part 1: the law of production, the value of a commodity
is determined (i.e., measured) by the amount of labour embodied in it. This
amount is measured in labour‑time units. (Here we ignore the reservations
made by Marx, according to whom labour = abstract homogeneous labour, and labour
time = time socially necessary for the production of a definite commodity etc.).
Consequently, value can be added to a definite product, for instance raw material,
only by adding labour.

2. according to the law of value, part 2: the law
of exchange, commodities are, on an average, sold at their value,
i.e., at prices corresponding to the amount of labour embodied in them. I.e.,
the exchange of commodities is an exchange of equivalents. 3. since labour
is conceived of as a commodity, its value is, according to the law of value,
part 1, determined by the amount of labour embodied in it. And according to
the law of value, part 2, it is, on an average, sold at its value.

4. Finally, since we are dealing with a capitalist
society, and apply the law of value to the relation between the labourer and
the industrial capitalist, it is that the capitalist sells his product with
a profit, but at its value.

However, these assumptions imply the contradiction
that the value of the commodity labour, under the same conditions, has two different
rnagnitudes. Marx's argument runs:

We assume that an industrial capitalist hires a
worker for the purpose of producing a definite commodity, and that he buys raw
material, machinery, tools, etc., on which and with which the labourer works,
at c price units; and "labour" from the labourer at v price
units, in the form of wages. According to Ricardo‑assumption 2. and
3., the value of raw material, machinery, etc. and "labour"
is c and v, respectively.

Having finished his work, the labourer has, according
to Ricardo‑assumption 1. added value to the raw material in an amount
of v; and the finished [21/22] product has, consequently, the
value c + v. Now, according to Ricardo‑assumption 2., the product
is sold at its value c + v. However, since, according to Ricardo‑assumption
4., it is sold with a profit p, it is sold at the value
c + v + p. Hence, the value of the capitalistically produced commodity
has two different magnitudes: c + v and c + v
+ p; or, the value of "labour" has two different
magnitudes: v and v + p.

At first sight, this paradox is destructive of the
labour theory of value, as far as capitalist society is concerned. And it would
seem that it imposes the task on the investigator to look for other possibilities
of explaining profit. However, in his analysis of the paradox, Marx does not
discover its roots in the premisses of the labour theory of value. He does not
draw the conclusion that the labour theory of value is inapplicable to capitalist
society, and that it, consequently, is unfit for the purpose of explaining profit.
Instead he looks for a deficiency in Ricardo's application of the theory; and
identifies the concept of "labour" as the stumbling‑block. In
particular, he concentrates on the Ricardo‑assumption that the total
labour of the labourer is paid for in the form of wages.

A clue to the solution of the problem is discovered
in the relationship between the feudal lord and the peasant serf, in pre‑capitalist
society. The peasant works, for example, three days for himself on his own field
or the field allotted to him, and three subsequent days he performs compulsory
and gratuitous labour on the estate of his lord. Here paid and unpaid parts
of labour are sensibly separated, separated in space and time`. Hence, if the
relationship between the industrial capitalist and the labourer involves a principal,
but invisible or hidden partition of the total labour performed, in labour paid
for and labour not paid for, the question to be answered is: can this invisible
partition be disclosed or established as a fact?

Since empirical evidence was hard to have, Marx
solved the problem in a purely logical way, through the construction of a consistent
conceptual system that explains profit, and through a reductio ad absurdum
argument. For this analysis the Ricardo‑paradox, purporting that the
value of "labour" has two different magnitudes, served
him as another clue.

Whereas it was quite natural to attempt a generalization
of the concept of commodity and the concept of value so as to comprize the "something"
which the labourer alienated on the market, the concept of "labour"
was obviously unfit for the purpose. Already in Grundrisse, Marx
suggested that labouring capacity" [98] might replace "labour"
in the description of the relationship between industrial capitalist and labourer.

This suggestion turned out to be of great advantage.
For if it is assumed that "labouring capacity" or labour power"
[99] is the commodity that the worker sells on the market at its value, the
way is cleared for the division of the expenditure of labour power into two
parts: one part paid for in the form [22/23] of wages, and
one part not paid for, that accrues to the industrial capitalist in the form
of profit. This, again, clears the way for the solution of the Ricardo‑paradox
through the generalization of the concept of commodity and the concept of value,
so as to comprize labour power. The description of the procedure of generalization,
known from classical mathematics and modern physics, is postponed to the presentation
of Marx's conception of conceptual development, that seems to be the genuine
part of Marx's dialectic.

Here it should only be added that Marx in his exposition
of the problem points out that "to explain. . . the general nature of
profits, you must start from the theorem that, on an average, commodities
are sold at their real values, and that profits are derived
from selling them at their values, that is, in proportion to the
quantity of labour realized in them. If you cannot explain, profit upon this
supposition, you cannot explain it at all. This seems paradox and contrary to
everyday observation. It is also paradox that the earth moves round the sun,
and that water consists of two highly inflammable gases. Scientific truth is
always paradox, if judged by everyday experience, which catches only the delusive
appearance of things" [100].

Here, again, Marx employs the Copernican paradigm,
meeting the requirement of consistency. In particular, the popular notion that
profits are derived from selling commodities over and above their
value is ruled out in a reductio ad absurdum argument, presented in Capital: "Suppose then, that by some inexplicable privilege, the seller is enabled
to sell his commodities above their value, what is worth 100 for 110, in which
case the price is nominally raised 10%. The seller therefore pockets a surplus
value of 10. But after he has sold he becomes a buyer. A third owner of commodities
comes to him as seller, who in his capacity also enjoys the privilege of selling
his commodities 10% too dear. Our friend gained 10 as a seller only to lose
it again as a buyer. The nett result is, that all owners of commodities sell
their goods to one another at 10% above their value, which comes precisely to
the same as if they sold them at their true value. . . Let us make the opposite
assumption, that the buyer has the privilege of purchasing commodities under
their value. In his case it is no longer necessary to bear in mind that he in
his turn will become a seller. He was so before he became a buyer; he has already
lost 10% in selling before he gained 10% as buyer. Everything is just as it
was".

Consequently, "the creation of surplus value
<or profit> . . . can be explained neither on the assumption that commodities
are sold above their value, nor that they are bought below their value"
[101]. Conclusion: profits can only he explained on the assumption that they
are sold at their value.

In order to be able to describe capitalist society "with
the precision of natural science" [102], Marx sets out to construct an
ideal capitalist society as sketched above. It comprizes two social classes,
the class of industrial capitalists and the class of wage‑workers. These
classes are related to each other in re-[23/24]lations of production
and exchange, to which the criterion of recurrence can be applied. The laws
governing these constant or recurrent relationships: the law of capitalist production
of commodities and the law of capitalist exchange of commodities,
are conveniently summarized in "the law of surplus‑value, distinct
from "the law of value" of theory I. "The law of surplus‑value"
is of course the basic law of theory II, the theory of capitalist society (in
the first approximation).

The analysis of the relationship between "the
law of value" and "the law of surplus-value" is postponed to
the presentation of Marx's conception of social development.

In the middle of the fifties, Marx realized that
"the precision of natural science" required not only the employment
of "the force of abstraction" and the method of idealization, but
a system of adequate symbols and mathematical expedients at that. In a letter
from January 11, 1858 [103], Marx informs Engels that he is engaged in a serious
study of mathematics, in order to be able to present his subject in an acceptable
scientific form. A rather elaborate set of symbols is already discovered in
a manuscript from February 1858. These symbols are connected in a primitive
system of equations, describing the relationships between surplus value, production
time, circulation time and turnover [104].

However, Marx was more ambitious than that. It would
seem that he planned an extension of the application of the calculus, developed
by Leibniz and Newton and employed by Newton in Principia, to
economics. In particular, he intended to give his "laws of motion"
the form of differential equations. In a letter from May 31, 1873, in a retrospect
on the situation, Marx informs Engels that he had attempted, several times,
in his analysis of crises, to determine mathematically the basic laws governing
"these ups and downs" [105]. Although Marx was convinced that the
project was feasible, in principle, he was dissuaded by Samuel Moore from continuing
his efforts, and the project was abandoned for the time being. Nevertheless,
he continued, almost till his death, the study of the differential and integral
calculus, and the analysis of the paradoxes to which it gave rise.

The outcome of these studies, which also cast an
interesting light on Marx's [106] conception of dialectics [106], was published
as Matematičeskie rukopisi in 1968. In our case these Mathematical
Manuscripts are important witnesses. Marx does not only quote Newton's Principia
(1687) and Arithmetica universalis (1707), but also other important
mathematical works from Euclid's Elements [107] to MacLaurin's A Treatise
of Algebra (1796) and Geometrica organica (1720) [107]. Marx's attempt
to furnish a mathematical formulation of the law, governing the movement of
the rate of profit, is a most interesting aspect of his endeavours to introduce
the precision of natural science into social science. The starting‑point
of these endeavours is of course the law of production and exchange [24/25]
of commodities of theory II, summarized as "the law of surplus value".
According to this law, part 1, the value of capitalistically produced commodities
is determined by the sum of constant capital (c), variable capital (v),
and surplus value (s), numerically equal to profit (p). According
to the same law, part 2, such commodities are on an average exchanged with other
commodities at their value. From this law, and from the definitions of
the rate of surplus value (s' = s / v), the rate of profit
(p’ = s / (c + v)), the organic composition of capital
(c / v), Marx derives the law of motion, p' = s’
/ ((c / v) + 1) [108].

The validity and/or applicability of this law has
been discussed since the first German edition of Capital, vol. III, was published in 1894. However, since the logical structure of the law hardly
was stated in a univocal way in Capital, a certain confusion concerning
its status has prevailed. This confusion culminated in K. R. Popper's The
Poverty of Historicism, where Marx is nominated "a famous historicist
" [109], on par with Hegel. Since this nomination totally ignores the impact
of mathematical and physical science on Marxian social science, basically distinct
from the influence it exerted on Hegel, it seems worth while to dwell on the
point at issue.

The questions at issue were: is the law, p'
= s’ / ((c / v) + 1) a universal law in the sense
that it refers to invariant or constant relationships in a capitalist society
as constructed and defined by Marx? Is it valid wherever and whenever
a capitalist society, as specified, comes into being? Or, is it a "hi­storical
law or trend", on the basis of which it might be possible to predict (or
prophesy) the impending break‑down of capitalism? I.e., was Marx a his­toricist
or an advocate of "the doctrine of historical laws or trends”? [110].

Admittedly, Marx did not succeed in formulating
the law of the movement of the rate of profit as a differential equation. He
did, however, formulate the law as an "equation . . . which . . . may also
be expressed by the proportion p':s' = v:(c + v);
the rate of profit is related to the rate of surplus‑value as the variable
capital is to the total capital" [111]. This is a conditional statement.
In Theorien über den Mehrwert Marx confirms that the law has this logical
structure, since it is presented as "the general law of the rise or
fall of the rate of profit" [112]. Obviously it does not follow from this
law "as such" that the rate of profit must fall. It may remain constant
or even rise, depending on the "values" of the variables c,
v, s.

In contradistinction to Hegel, Marx was well aware
that nothing follows from a law "as such"; that a natural or social
law formulated in mathematical terms is completely silent concerning the future,
that only the determination [25/26] of some initial state lends it voice;
that, in principle, any arbitrary initial state is possible. Being familiar
with classical mathematics and physics, he understood that one cannot determine
such initial state, unless one looks for observational or statistical "values"
of the variables in question.

However, prediction of social events was
not Marx's main concern. He was aware that the predictive power of the law of
the movement of the rate of profit, as formulated, was short‑ranged and
poor. He wanted above all to explain social facts, events or processes,
appearing on the surface of capitalist society. Hence, since he looked for an
explanation of definite social facts or processes, he was obliged to
look for or discover a law of motion (and not a historical law), from which
the statements describing social facts or processes could be derived. And that
was precisely what he did.

On the basis of "the law of surplus‑value"
profits were explained. On the basis of the law, p' = s’
/ ((c / v) + 1), the actual tendency of the rate of profit
was explained, that is, it was shown that the known "values"
of the variables c, v, s, agreed with the actual "value"
of p’. Accordingly, "the hypothetical series drawn up at the beginning
of this chapter <i.e., chapter XIII: "The law as such"> expresses.
. . the actual tendency of capitalist production" [113].

Moreover, Marx did not discover the fact
that, in classical capitalist society, the rate of profit had a tendency
to fall. Adam Smith dealt with the falling rate of profit some 75 years before
Marx came across the phenomenon and explained it. In The Wealth of
Nations [114]. Adam Smith quotes the long‑term interest rates in Great
Britain and in other countries as evidence of the declining tendency of the
rate of profit. In Capital Marx mentions "the concern of the English
economists over the decline of the rate of profit", adding: "The fact
that the bare possibility of this happening should worry Ricardo, shows his
profound understanding of the conditions of capitalist production" [115].

If this is so, what is the root of the confusion
concerning the logical status of the law governing the movement of the rate
of profit? What is wrong with the discussion concerning the validity and/or
applicability of the law? It would seem that the trouble originates in an ambiguous
use of the term "law" in Capital. In a letter to Engels, April 30,
1868, Marx mentions "the tendency of the rate of profit to fall as societyprogresses". This is a categorial statement, a statement of fact,
a description of a process. But it is certainly not a law. However,
it follows from the law stated above, plus definite "values"
of the variables c, v, s. This is confirmed by Marx in
the same letter, where he adds: "This already follows from what was developed
in Book I <i.e., Capital, vol. I> on the change in composition
of capital with the development of the social productive forces. This is
one of the greatest triumphs over the great pons asini of all previous
economics" [116]. [26/27]

Nevertheless, the title of Part III of Capital,
vol. III, is: "The law of the tendency of the rate of profit to fall"
[117]. This oddity is possibly a remnant of Hegel's terminology and conception
of the historical process, and may be justifiable if one insists on distinguishing
between "universal law", exemplified by the above equation, and "empirical
law", exemplified by the above categorical statement of fact. Whatever
modus loquendi one prefers, such "empirical law" cannot possibly
be a "law of motion" in the Marxian sense.

Be that as it may: if the phrase, "The law
of the tendency of the rate of profit to fall", is abandoned, Popper's
invectives against Marx's conception of explanation and prediction in social
science are without object; and his paraphrase on the XI. thesis on Feuerbach:
"The historicist can only interpret social development and aid it
in various ways; his point, however, is that nobody can change it"
[118], is not so funny as it might seem at first sight.

Since the laws of motion of capitalist society in
Marx refer to constant or "necessary" relationships, viz. the relation
of production and exchange, the change of that society can only take place within
these relations. I.e., it is a change of its "contingent " or "arbitrary"
elements. These elements are envisaged in the variables c, v,
s, of the equation, whose "values" define a definite "state"
of the social system constructed by Marx.

When Marx in the "Preface" to the first
edition of Capital deals with "the natural laws of capitalist production",
and refers to "tendencies working with iron necessity towards inevitable
results", it does not mean, as Popper seems to believe, that the laws "produce"
the same "inevitable results", regardless of what human beings are
doing. It only means that the law "makes its way", whatever initial
states of the social system are established. The point is, however, that these
initial states are created, or chosen, through economic and/or political activities
of the members of the relations of production and exchange; and the "inevitable
results" are as different as the initial states chosen are different. I.e.,
the rate of profit may fall, rise or remain constant, depending on how economic
activity and class struggle influence the organic composition of capital and
the rate of surplus‑value.

In the discussion of explanation and prediction
in Marx, "laws of motion", “statements of fact", "historical
laws", "laws of development", "dialectical laws", political
slogans, etc. are often mixed up in an unpalatable maze. Popper's views are
exemplary in this respect [119]. In Marx development proper is defined as the
transition from one social formation into another [120]. And in accordance with
the method of abstraction it is "disregarded" or "left aside"
for the time being. The problem may conveniently be postponed to our sketch
of the method of presentation.

In order to shed some light on Marx's thought-experimental
method, another example of explanation of social phenomena according to the
Coper-[27/28]nican appearance/essence pattern should be presented. It
was already mentioned above [121].

As we know, the object of analysis in Capital,
vol. I, is an extremely simplified and elementary capitalist society,
whose essential constituents are the class of industrial capitalists and the
class of wage‑workers. Landowners, money‑lenders, merchants as separate
classes are "thought away" in this society. "We treat the capitalist
producer as owner of the entire surplus‑value, or, better perhaps, as
the representative of all the sharers with him in the booty" [122].

This procedure has the obvious advantage that the
production of surplus‑value can be investigated independently of
the distribution of surplus‑value among landowners and capitalists
of all descriptions. One is inclined to believe Marx, who in a letter from August
24, 1867, announces that "the treatment of surplus‑value independently
of its particular forms as profits, interest, ground‑rent, etc."
[123], is one of the best points in Capital. In fact, it makes
a reduction of the number of variables to a manageable minumum possible; and
makes a comprehensive thought experiment with these variables feasible. We already
know that these variables: c or v or s, are elements in
the definition of the value of capitalistically produced commodities, of the
value of labour‑power, of the rate of surplus‑value, of the rate
of profit, of the organic composition of capital, of the productivity of labour,
etc. The application of the thought‑experimental procedure to these variables
is the basis of the above definitions and laws of motion established. And finally:
on the basis of the laws of motion "the apparent movement is explained"
[124], that is, the apparent sources of income: "capital 
profit (profit of enterprise plus interest), landground‑rent,
labourwages". Or, "the trinity formula which comprises all
secrets of the social production process" [125] is explained.

Although Kant does not coin the phrase "thought
experiment", he points out that an essential part of the business of research
"consists in analysis of the concepts we already have of objects"
[126]. And he strongly urges that constructive activity and operation
in thought with the objects or quantities involved is another essential part
of the act of cognition, as far as mathematics and physics are concerned"
[127]. We have already quoted illuminating examples of such procedures in Galileo
and Newton. It would even seem that Newton, who often is accused of "inductionism",
is a most outstanding exponent of the thought‑experimental method. The
reader of Principia cannot help perceiving this. Certainly, Kant did
not overlook this aspect of Newtonian mathematical physics.

It is quite typical of the propositions presented
by Newton in Principia to have the form of problems. This
is well illustrated by Section III of Book I: "The motion of the bodies
in eccentric conic sections", where the propositions have the form: "If
a body revolves in an X; it is required to find the law of the [28/29]
centripetal force tending to the focus of that figure". Or, "Suppose
a body to move etc. . .". For X one can of course substitute "ellipse",
"hyperbola", or "parabola". At the end of the argument,
one does not find the Q.E.D. (Quod erat demonstrandum), but the Q.E.I. (Quod
erat inveniendum), that is: "Which was to be invented" [128]. In other cases the argument ends with a Q.E.F. (Quod erat faciendum), that
is: "Which was to be done or constructed" [129]. Book I
and II of the Principia are abundant in examples like that.

There is hardly a scrap of textual evidence
in favour of the conception that Marx took over an essential part of his thought‑experimental
procedure from Newton's Principia. If we turn the pages of Mathematical
Manuscripts, we discover that Marx mostly concentrates on logical
problems concerning the differential calculus. In his rough sketch of the historical
development of the calculus Marx quotes the Scholion to Lemma XI of Book I,
and Lemma II of Book II of the Principia [130], dealing with the concepts
of limit and momentum in Newton. Otherwise Marx is interested
in conic sections. But he does not refer to Newton's thought experiments with
conic sections in Principia. Marx's knowledge of the subject stems, it
would seem, above all from Sauri's Cours complet de mathématiques (1778)
[131]. Nevertheless, we can be sure that Marx was familiar with methods used
in classical mathematical physics before he wrote Capital, and that he
acquired his knowledge in this field, not through the study of Hegel's Logik
and Naturphilosophie, but through inquiry into the best authorities
in the field of mathematics and physics from Euclid and Boëtius to Newton, Taylor
and McLaurin, and from Euler to d'Alembert, Lagrange and Laplace [132]. It is
remarkable that Marx in a short sketch of his method of investigation, observes
that "analysis . . . first is carried on purely in the mathematical field"
[133]. One cannot help calling to mind Newton's Preface to the first edition
of Principia, where he informs the reader: ". . . by the propositions
mathematically demonstrated in the former Books, in the third I derive from
celestial phenomena the forces of gravity with which the bodies tend to the
sun and the several planets.” [134]

Of course it is outside the scope of this paper
to treat of Marx's thought experimental method in detail. It may suffice to
point to the way in which he arrived at some basic concepts, definitions or
classifications that served as a foundation for the formulation of laws of motion
in mathematical terms, and as a basis for explanation of "the apparent
movement".

We have already mentioned the purely analytic thought
experiment with the Ricardian concept of "labour", and the subsequent
construction of the concept of labour‑power. Within the framework of the
capitalist society presented in Capital, vol. I, several constructive
thought experiments are performed, which typically begin with propositions of
the form, "Let us assume that such and such. . .", "If such and
such remains unchanged. . .", "In this [29/30] chapter such
and such is a constant magnitude . . .". Suppositions of this type make
possible the "variation in thought" of definite quantities or "factors",
while others remain unchanged.

The basic assumption is that capitalist society
consists of two classes, the class of industrial capitalists and the class of
wage‑workers. Consequently, the analysis deals solely with the production
of surplus‑value; and surplus‑value is, on this stage, numerically
identical with profit. This supposition is valid only until the conversion of
profit into average profit is introduced, and the distribution of surplus‑value
is taken into consideration [135].

The analysis of this simplified capitalist society
starts in Part III of Capital, "The production of absolute surplus‑value",
with the assumption that the mode of production is given and invariable, that is, that the productivity of labour is given and invariable; and that, consequently, "the value of labour‑power, and therefore
the part of the working‑day necessary for the reproduction or maintenance
of that labour‑power are. . . given constant magnitudes" (italics
mine). In the first phase of the thought experiment also the working‑day
is a given constant magnitude [136].

Within this framework, the method of calculation
of the rate of surplus‑value is presented: "We take the total value
of the product <viz. c + v + s> and put the constant
capital which merely re‑appears in it, equal to zero. What
remains, is the only value that has, in the process of producing the commodity,
been actually created. If the amount of surplus‑value be given, we have
only to deduct it from the remainder, to find the variable capital. And vice
versa, if the latter be given, and we require to find the surplus‑value.
If both be given, we have only to perform the concluding operation, viz. to
calculate s / v, the ratio of surplus‑value to the variable
capital" [137].

In the next phase of the experiment, the condition
that the working‑day is a given constant magnitude, is suspended; and
the assumption is made that the mass of surplus‑value is increased through
the prolongation of the working‑day. Surplus‑value produced by the
prolongation of the working‑day is called absolute surplus‑value
[138].

In Part IV of Capital, "Production
of relative surplus‑value" the condition that the productivity of
labour is given and invariable, is suspended, and the investigation continues
with the assumption that the productivity of labour increases. This involves
an alteration in the labour‑process, of such a kind as to shorten the
labour‑time socially necessary for the production of a commodity.

It is now possible, with reference to "the
law of surplus‑value", to define "relative surplus‑value"
as surplus‑value arising from the curtailment of the necessary labour‑time,
and to formulate the following law: "The value of com­modities is in the
inverse ratio to the productiveness of labour" [139]. This law, again,
requires of course a definition of the productiveness of labour, and a mo[30/31]­dification
in its employment to the commodity labour‑power [140]. However, this problem
should not concern us here.

Until now, the relationship between constant capital
(c) and variable capital (v), that is, the organic composition
of capital (c / v), remained unchanged. The actual conditions
under which such change takes place, is investigated in Capital, vol.
I, under the heading, "The general law of capitalist accumulation"
[141]. Here, where we are concerned with the analysis "purely in the mathematical
field", we pass over the investigation of the actual con­ditions
under which (c / v) changes. Therefore, the so‑called "general
law of capitalist accumulation" is left out of consideration, so much the
more as the "law" is a statement of fact or an "empirical
law", dealing with the actual situation in capitalist Great Britain
in the sixties, rather than a "law of motion" in the Marxian sense.

Marx continues his thought‑experimental procedure,
aided by elementary mathematical symbols and expedients, with the definition
of the rate of profit, p’ = s / (c + v), and the
derivation of the universal law, p' = s’ / ((c / v)
+ 1). From there on, he makes a series of assumptions concerning constancy or
variation of the quantities involved. Under the supposition, "s',
v and (c + v) variable" [142], live cases are enumerated:
"It follows from all of these five cases. . . that a rising rate of profit
may correspond to a falling or rising rate of surplus‑value, a failing
rate of profit to a rising or falling rate of surplus‑value, and a constant
rate of profit to a rising or falling rate of surplus‑value. And we have
seen in I that a rising, failing, or constant rate of profit may also
accord with a constant rate of surplus-­value" [143].

Here, again, we leave out of consideration Marx's
application of the law to contemporary capitalist society. We know that, if
the situation is such that "it involves changes in the average organic
composition of the total capital of a certain society, then the gradual growth
of constant capital in relation to variable capital must necessarily lead to
a gradual fall of the general rate of profit, so long as the rate
of surplus‑value, or the intensity of exploitation of labour by capital,
remain the same" [144]. In the conviction that these conditions will be
fulfilled throughout the lifetime of capitalism, Marx calls the statement concerning
the gradual fall of the general rate of profit, a "law of capitalist production"
[145], a modus loquendi that gives rise to the ambiguity and confusion
mentioned above.

In his thought experiment, Marx continues his investigation
of a simplified capitalist society, comprizing two social classes, the class
of industrial capitalists and the class of wage‑workers. The purpose is
to analyze the production of surplus‑value, as independent of the distribution
of surplus‑value. On this [31/32] stage surplus‑value is
still numerically equal to profit, and the rate of profit may rise or fall,
while the rate of surplus‑value remains constant. In the following analysis
it is assumed that the rate of surplus‑value is the same in all spheres
of production competing with each other, and that the length of the working‑day
remains constant. And of course the whole reasoning is based on "the law
of surplus‑value", part 2, according to which all commodities are
sold at their value.

However, precisely this assumption leads to a surprising
paradox, that apparently is destructive of the labour theory of value, as far
as competitive capitalism is concerned.

According to the presuppositions enumerated, "different
lines of industry have different rates of profit, which correspond
to differences in the organic composition of their capitals and, within
indicated limits, also to their different periods of turnover; given the same
time of turnover, the law (as a general tendency) that profits are related to
one another as the magnitudes of the capitals, and that, consequently, capitals
of equal magnitudes yield equal profits in equal periods, applies
only to capitals of the same organic composition (c / v),
even with the same rate of surplus‑value" (italics mine)" [146].

As it turns out, a new stumbling‑block to
the labour‑theory of value comes to light: "There is no doubt, on
the other hand, that aside from unessential, incidental and mutually compensating
distinctions, differences in the average rate of profit in the various branches
of industry do not exist in reality, and could not exist without
abolishing the entire system of capitalist production" [147]. In modem
parlance: The labour‑theory of value is falsified.

Marx takes stock of the
situation in the following words: "It would seem, therefore, that here
the theory of value is incompatible with the actual process, incompatible with
the real phenomenon of production, and that for this reason any attempt to understand
these phenomena should be given up" [148].

As we know, the attempt to explain "the apparent
movement" or the distribution of surplus‑value among industrial capitalists,
merchants, money‑lenders and landlords, is not given up. In order to reach
this goal, Marx continues analysis, and makes use of a device known in classical
mathematics and, in particular, modem physics: the procedure of "rational
generalization". This means that the concept of value is generalized so
as to comprize the "new" phenomena, characteristic of competitive
capitalism. This is done through the construction of the concept of "price
of production". It is defined in terms of "average rate of profit".

"The prices which obtain as the average of
the various rates of profit in the different spheres of production added to
the cost‑prices of the different spheres of production, constitute the
prices of production. They have as their prerequisite the existence of
a general rate of profit, and this, again, presup-[32/33]poses that the
rates of profit in every individual sphere of production taken by itself have
previously been reduced to just as many average rates. These particular rates
of profit are equal to s / (c + v) in every sphere of production,
and must, as occurs in Part I of this book <i.e., Capital, vol. I>,
be deduced out of the values of the commodities" [149]. This is the famous
transformation problem [150], still subject to controversy. However,
as Marx points out: "Without such deduction the general rate of profit
(and consequently the price of production of commodities) remains a vague and
senseless conception". Hence, the prim of production of a commodity is
equal to its cost‑price <k = c + v> plus the profit allotted
to it in per cent, in accordance with the general rate of profit (i.e., kp'>,
or, in other words, to its cost‑price plus the average profit"[151]
(i.e. k + kp'av = c + v + pav>,
where k is the cost‑price, p’av the average rate of profit,
and pav the average profit.

Since commodities under competitive capitalism are
sold, not at their values, i.e., at c + v + p (where p
= s), but at their prices ofproduction, i.e., at c
+ v + pav, it is obvious that, under these
conditions, surplus‑value is no longer numerically equal to profit. Consequently,
the law of the movement of the rate of profit does not refer plainly to profit
(p), but to average profit (pav). Furthermore,
according to the new law of exchange, market‑prices do not oscillate around
values of commodities, but around their prices of production [152].

As far as the production of surplus‑value
is concerned, we have arrived at the highest stage of complexity; and although
not all problems are solved in an univocal way, it is now possible, in principle,
to solve the distribution or division problem. "We intentionally
present this law (p'av = s’ / ((c
/ v) + 1).) before going on to the division of profit into different
independent categories", namely industrial profit, commercial profit, interest
and ground‑rent. "The fact that this analysis is made independently
of the division of profit into different parts, which fall to the share of the
different categories of people", viz. industrial capitalists, merchants,
money‑lenders and landowners, "shows from the outset that this law
is, in its entirety, independent of this division, and just as independent of
the mutual relations of the resultant categories of profit” [153].

As we perceive, Marx now takes leave of the very
simplified capitalist society whose relations of production primarily became
the object of analysis proper, and re‑introduces "the most popular,
and so to say, antediluvian forms, merchants' capital and money­lenders' capital",
that were entirely left out of consideration at the beginning of this analysis.
From now on the competition of capitalists is no longer restricted to the rivalry
of industrial capitalists, but is extended so as to comprize merchants, money‑lenders
and [33/34] landowners. This is, of course, a step into still greater
complexity. Here we e we refrain from going into details, and restrict ourselves
to reference to parts IV‑VII of Capital, vol. III [154].

However, once again, we want to call attention to
the impact of classical mechanics on Marxian social science. As intimated above,
this dependence is particularly conspicuous in the attempt to solve the problem
of distribution or division of surplus‑value, which at the end of Capital, vol. III, is reworded as the problem of "revenues and their sources"
[155]. Pointing out the contradictions or absurdities that serve "vulgar
economy" as premisses, Marx makes the following illuminating remark: "It
should not astonish us, then, that vulgar economy feels particularly at home
in the estranged outward appearances of economic relations in which these
prima facie absurd and perfect contradictions appear and that these relations
seem the more self‑evident the more their internal relationships are
concealed from it, although they are understandable to the popular mind"
(italics mine).

Referring to the classical (Copernican) paradigm,
he continues: "But all science would be superfluous if the outward appearance
and the essence of things directly coincided . . . Thus, vulgar economy has
not the slightest suspicion that the trinity which it takes as its point of
departure, namely, land‑rent, capital‑interest, labour‑wages
or the price of labour, are prima facie three impossible combinations.
First we have the use‑value land, which has no value, and the exchange‑value
rent: so that a social relation conceived as a thing is made proportional
to Nature, i.e., two incommensurable magnitudes are supposed to stand in a given
ratio to one another. Then capital‑interest. If capital is conceived
as a certain sum of values represented independently by money, then it is prima
facie nonsense to say that a certain value should be worth more than it
is worth. It is precisely in the form: capital‑interest that all intermediate
links are eliminated, and capital is reduced to its most general formula, which
therefore in itself is also inexplicable and absurd. The vulgar economist prefers
the formula capital‑interest, with its occult quality of making a value
unequal to itself, to the formula capital‑profit, precisely for the reason
that this already more nearly approaches actual capitalist relations" [156].

In the traditional discussion of Marx's contribution
to the development of methods in political economy and sociology, the impact
of classical mathematics and physics on Marxian social science was heavily neglected.
Even Mathematical Manuscripts, published in 1968, did not attract the
attention of philosophers and sociologists in the way they deserved. On the
other hand, the revived interest in Okonomisch-philosophische Manuskripte
(1843), Grundrisse and other works from the forties and fifties,
associated with renewed Hegel‑studies, led to an excessive overestimation
of Marx's dependence on Hegel. [34/35]

For a due evaluation of Marxian social science,
this was extremely harmful, and caused a long‑winded and futile discussion
in worn‑out terms of the materialist conception of history and society
and of Marxian dialectics.

According to our analysis, "the method of dialectic"
consists roughly of two parts. One part comprizes conceptions, procedures and
devices, practically identical with those developed in physics since the time
of Copernicus. Their elements were enumerated or depicted above. Another part
stems undoubtedly from Hegel. For one thing, Hegel, following in Kant's footsteps,
observes in his Encyclopädie that "the main point to be noted is
that antinomies are inherent, not only in the four special objects taken from
cosmology, they are rather inherent in all objects pertaining to all
classes, in all representations, concepts and ideas. To know this and
to take cognizance of the objects from this point of view, is an essential part
of the philosophical conception; it is the core of that which, further, turns
out to be the dialectical aspect of the subject of logic" [157].

This view was, with a certain proviso, accepted
by Marx, and associated with his conception of conceptual and historical development,
involving the discovery and solution of interesting paradoxes. The most illuminating
example of such change is presented in Capital, vol. I, Part II:
"The transformation of money into capital" [158]. Another example
is discovered in Capital, vol. III, chapter IX: "Formation of a
general rate of profit (average rate of profit) and transformation of the values
of commodities into prices of production" [159].

Since, for trivial reasons, the present essay could
not be finished according to schedule, it was not possible, here, to present
Marx's conception of conceptual development, provided for in the initial plan,
and held out as prospect on p. 23. Nor was it feasable to establish its relationship
to the procedure of' generalization, known from classical mathematics and modern
physics. However, in a paper, "Reflections on Marxian Dialectics"
[160], a rather detailed account of Marx's description of a process of conceptual
development, and its relationship to the succession in history of definite relations
of production, was furnished. And some suggestions were given for further study
of the relationship between a dialectical process and "mathematical generalization",
as employed in Bohr's correspondence argument [161].

Hegel, once again following in Kant's footsteps,
expounded the view that the process of knowledge or cognition is a productive
act. Also this view was accepted by Marx, with the important modification or
addition that "the production of ideas, of conceptions, of consciousness,
is at first directly interwoven with the material activity and the material
intercourse of men, the language of real life. Conceiving, thinking, the mental
intercourse of men, appear at this stage as the direct efflux of their material
behaviour. The same applied to mental production as expressed in the language
of politics, laws, [35/36] morality, religion, metaphysics, etc., of
a people. Men are the producers of their conceptions, ideas, etc.real,
active men, as they are conditioned by a definite development of their productive
forces and of the intercourse corresponding to these, up to its furthest forms”
[162].

However, Marx never developed, in detail, the analogy
between mental and material production which he discovered, and did not succeed
in creating a conceptual system adequate for the description of mental production
and communication. Although we all over in Capital and Theorien über
den Mehrwert discover valuable hints and suggestions, relevant to the problem,
Marx was unable to raise a theoretical structure connecting these two fundamental
spheres of human activity. It would seem that Marx's most important contribution
to the solution of the problem was the discovery of paradoxes that come into
being, if one imagines that mental products, for instance "the law of deviation
of the magnetic needle in the field of an electric current" [163], or the
binomial theorem [164], are taken to the market for exchange as commodities.

Since the industrial capitalist, the wage‑labourer
and the producer of scientific knowledge have key‑positions in capitalist
society, they are mutually connected in social relations, characteristic of
this society. The social relation between the industrial capitalist and the
producer of scientific knowledge is so far similar to the social relation between
the industrial capitalist and the labourer, as "the science of others is
as much annexed by capital as the labour of others" [165]. This is so,
although neither labour-­power nor scientific knowledge are produced for the
market with a view to sale.

Nevertheless, the appropriated labour‑power
has a social status, quite different from the social status of the appropriated
scientific knowledge. For whereas the labour‑time socially necessary for
the production and reproduction of labour‑power, and consequently, its
costs of production and reproduction, can be properly determined and measured,
the labour‑time necessary for the reproduction of the products of mental
labour is in no keeping with the labour‑time required for its original
production. For instance, "a schoolboy may learn the binomial theorem in
one hour" [166]. That is, whereas the costs of production of scientific
knowledge may be substantial, a law or a theorem, once discovered, cost never
a penny [167]. This goes even so far that "science, generally speaking,
costs the capitalist nothing, a fact that by no means hinders him from exploiting
it" [168]. It would therefore seem that scientific knowledge, although
it has a use‑value, in as far as it is technologically exploitable,
has no value in the Marxian sense.

This is very queer, indeed! For since scientific
knowledge, for its production, requires part and parcel of the total, socially
necessary, labour‑time of society, it should, according to the labour‑theory
of value, have a definite value.

Here, again, we come across
a paradox or a stumbling‑block to the labour‑theory of value; and
it suggests itself to ask the question: what is wrong with [36/37] the
analogy between material and mental production, discovered by Marx? How far
does this analogy take us? Does the paradox indicate a new possibility of generalizing
the labour‑theory of value?

If scientific knowledge were a means of production,
i.e., an instrument of labour, raw material, or an auxiliary substance, it should,
in the process of production, transfer its value to the new product, thereby
gradually reducing its own value to zero. However, in contradistinction
to a means of production, scientific knowledge can be used again and
again in the process of production without being worn down. Furthermore, since
scientific knowledge, in contradistinction to labour‑power, is not a source
of value either, it does not transfer or add value to the new product in the
process of production. Its role is different. It raises productivity.

This is particularly conspicuous in competitive
capitalism, where "know-how", recipes or licences, that are products
of "industrial research", are offered for sale on the market and bought
by industrial capitalists with a view to securing a so‑called "extra‑profit".
Under these circumstances the paradox is accentuated. For whereas the
costs of production, within the Marxian regime, are measured in units of value,
the exchange value of the product is measured in units of productivity.

This seems to be the end of the analysis of the
problem as far as Marx is concerned. Since the problem, as posited by
Marx, remained unsolved, it wits not possible to reintroduce mental production
and the corresponding relations of production, left out of consideration at
the beginning of the analysis, into the theoretical structure presented in Capital.
Therefore, also in this respect, Marx's work remained a torso.

Marx was a classic in more
than one respect. Inspired by the "Scottish Historical School", and
aided by his own studies of classical mathematics and physics, he introduced
and employed methods of investigation and presentation that enabled him to complete
the work begun by Adam Smith and Ricardo. However, even these methods, which
completely outstripped Hegelian dialectics, were hardly adequate for the solution
of all the problems posited and formulated by Marx, not to speak of the problems
which came into being a century later.

72 MESC, p. 109
(Marx to Engels, April 2, 1858). "Money, as the development of its
determination shows, contains within itself the demand for value which will
enter into circulation, maintain itself during the circulation, and at the same
time establish circulation  that is, for capital. This
transition is historical also".

73 R Meek, Economics and . . ., op.
cit., pp. 97‑98.

74 K Marx, Capital, vol. I, op. cit.,
p. 100.

75 Ibid., p. 129.

76 Ibid., pp. 42‑162.

77 See: p. 1.

78 K Marx. A Contribution to the Critique of
Political Economy, op. cit., p. 187.

SOURCE: Witt-Hansen, Johannes. "Marx's Method in Social Science,
and Its Relationship to Classical and Modern Physics and Mathematics",
Poznan Studies in the Philosophy of the Sciences and the Humanities,
vol. 3, nos. 1-4, 1977 (Amsterdam: B.R. Gruner Publishing Co.), pp. 1-41. (Issue
theme: Aspects of the Production of Scientific Knowledge, edited by J.
Witt-Hansen.)